Thixoforming : Semi-solid Metal Processing

378j 10 Thixoforging and Rheoforging of Steel and Aluminium Alloys
Introduction to Flatness-based Control
Flatness is a property of a control system which allows us to trivialize the trajectory
planning tasks for the open-loop control, without solving differential equations. The
curved coordinates of a nonlinear flat system can be converted into flat coordinates,
similar to the coordinates of a linear system [31]. If a system is flat, we can directly
express the states and inputs, without integrating any differential equation, in terms
of the so-called flat output and a finite number of its derivatives [32]. More precisely, if
the system has states x and inputs u then the system is flat if we can find outputs y of
the form [33]
y ¼ h x; u; _u; ...; u r
such that
x ¼ j y; _y; ...; y q
and
u ¼ a y; _y; ...; y
qþ1
ð10:9Þ
The number of inputs has to be equal to the number of outputs (dim y ¼ dim u). If
the system is flat, we can calculate the inputs with respect to the desired outputs
which guarantee that the outputs follow the desired behaviour. The desired
trajectories for the outputs y have to be functions which are (q þ 1)-times
differentiable. The trajectories can be expressed by polynomials which are sufficiently
often differentiable [31]. If a system is flat, it is an indication that the
nonlinear structure of the system is well characterized and one can exploit that
structure in designing control algorithms for motion planning, trajectory generation
and stabilization [33]. There are numerous applications of flatness to systems
with concentrated parameters [32] and lately to distributed systems also. Flatnessbased
control is therefore chosen to control the inductive heating and the forming
process.
Examples of flatness-based control for heat transfer problems can be found
in [34–37]. These systems are boundary controlled and the material properties are
constant, that is, the system is influenced by a controlling element at the systems
boundary and the material properties are not temperature dependent. The problem
with inductive heating is that the system has a distributed input, the volume power
density _Fðr; tÞ and, especially for the inductive heating of steel, the material
properties are highly temperature dependent. Hence the open-loop control schemes
mentioned previously cannot be applied to the given problem. Therefore, a new
approach for feed-forward control with the finite difference method is explained in
the following.
Flatness-based Control of Inductive Heating
Depending on the model of the heat transfer, there are two methods to determine a
flatness-based control. One method was applied to the inductive heating of aluminium.
This approach can be found in [38] and will not be described here because the